The invention concerns the digital equalization of signals. It has important applications in the field of radio communications.
The method is applied to the processing of a signal received from a transmitter over a channel between transmitter and receiver, whose response is known or has been estimated beforehand. One of the main issues is then that of the compromise between the performance of the equalizer and its complexity.
A complete maximum likelihood estimation of the transmitted signal is possible, for example by employing the Viterbi algorithm (see G. D. Forney Jr.: The Viterbi Algorithm, Proc. of the IEEE, Vol. 61, No. 3, March 1973, pages 268-278). Nevertheless, as soon as the impulse response of the channels becomes long or the number of possible discrete values becomes large, the exponentially increasing complexity of these methods renders them impracticable.
We consider the case of a radio communications channel used for the transmission of a signal composed of successive sequences or frames of n symbols, dk (1≦k≦n). The symbols dk have discrete values: binary (±1) in the case of a BPSK (binary phase shift keying) modulation; quaternary (±1+j) in the case of a QPSK (quaternary phase shift keying) modulation.
After baseband conversion, digitization and matched filtering, a vector Y of the received signal, corresponding to the symbols transmitted over the duration of a frame, is defined by the expression                     Y        =                              (                                                   ⁢                                                                                y                    1                                                                                                                    y                    2                                                                                                                                                                                                                     ⋮                                                                                                                                                                                                                       y                    k                                                                                                                                                                                                                     ⋮                                                                                                                                                                                                                       y                    L                                                                        ⁢                                                   )                    =                                                                      (                                                                           ⁢                                                                                                              r                          0                                                                                            0                                                                    0                                                                    ⋯                                                                    0                                                                                                                                      r                          1                                                                                                                      r                          0                                                                                            0                                                                                                                                                                                           ⋮                                                                                                                                                                                                                                                             r                          1                                                                                                                      r                          0                                                                                            ⋰                                                                    0                                                                                                            ⋮                                                                                                                                                                                                                     r                          1                                                                                            ⋰                                                                    0                                                                                                                                                                                                                                   ⋮                                                                                                                                                                                           ⋰                                                                                              r                          0                                                                                                                                                              r                          w                                                                                                                                                                                                                   ⋮                                                                                                                                                                                                                     r                          1                                                                                                                                    0                                                                                              r                          w                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         0                                                                    0                                                                                              r                          w                                                                                                                                                                                                                   ⋮                                                                                                            ⋮                                                                                                                                                                                           ⋰                                                                    ⋰                                                                                                                                                                                                                                   0                                                                    ⋯                                                                    0                                                                    0                                                                                              r                          W                                                                                                      ⁢                                                                           )                                ⁢                                  (                                                                           ⁢                                                                                                              d                          1                                                                                                                                                              d                          2                                                                                                                                                                                                                                                                                                   ⋮                                                                                                                                                                                                                                                                                                     d                          k                                                                                                                                                                                                                                                                                                   ⋮                                                                                                                                                                                                                                                                                                     d                          n                                                                                                      ⁢                                                                           )                                            +                              (                                                                   ⁢                                                                                                    y                                                  N                          ,                          1                                                                                                                                                                        y                                                  N                          ,                          2                                                                                                                                                                                                                                                                                                 ⋮                                                                                                                                                                                                                                                                           y                                                  N                          ,                          k                                                                                                                                                                                                                                                                                                 ⋮                                                                                                                                                                                                                                                                           y                                                  N                          ,                          L                                                                                                                    ⁢                                                                   )                                      =                                          A                .                D                            +                              Y                N                                                                        (        1        )            where W+1 is the length in numbers of bits of the estimated impulse response of the channel, r=(r0, r1, . . . , rw) is the estimated impulse response of the channel, the rq being complex numbers such that rq=0 for q<0 or q>W, Yk is the k-th complex sample received with 1≦k≦L=n+W, and YN is a vector of dimension L composed of samples of additive noise yN,k. The estimated impulse response r takes into account the propagation channel, the signal shaping by the transmitter and the receiver filtering.
The matrix A of dimension L×n has a Toeplitz-type structure, meaning that if αi,j denotes the term in the i-th row and in the j-th column of the matrix A, then αi+1,j+1=αi,j for 1≦i≦L−1 and 1≦i≦n−1. The terms of the matrix A are given by: α1,j=0 for 1<j≦n (A therefore has only zeros above its principal diagonal); αi,1=0 for W+1>i≦L (band-matrix structure); and αi,1=ri−1, for 1≦i≦W+1.
The matrix equation (1) expresses the fact that the signal received Y is an observation, with an additive noise, of the convolution product between the channel impulse response and the transmitted symbols. This convolution product can also be expressed by its Z-transform:Y(Z)=R(Z).D(Z)+YN(Z)  (2) where D(Z), Y(Z), R(Z) and YN(Z) are the Z-transforms of the transmitted symbols, of the received signal, of the impulse response and of the noise, respectively:                               D          ⁡                      (            Z            )                          =                              ∑                          k              =              1                        n                    ⁢                                           ⁢                                    d              k                        ·                          Z                              -                k                                                                        (        3        )                                          Y          ⁡                      (            Z            )                          =                              ∑                          k              =              1                        L                    ⁢                                           ⁢                                    y              k                        ·                          Z                              -                k                                                                        (        4        )                                          R          ⁡                      (            Z            )                          =                              ∑                          q              =              0                        W                    ⁢                                           ⁢                                    r              q                        ·                          Z                              -                q                                                                        (        5        )            
A conventional solution to solve a system such as (1) is the so-called “zero forcing” method, by which we determine the vector {circumflex over (D)}ZF of n continuous components which minimizes the quadratic error ε=∥A{circumflex over (D)}−Y∥2. Subsequently, a discretization of the components of vector {circumflex over (D)}ZF relating to each channel is performed, often through a channel decoder. The least-squares solution {circumflex over (D)}ZF is given by {circumflex over (D)}ZF=(AHA)−1AHY, where AH denotes the conjugate transpose of matrix A. We then are faced with the problem of inverting the Hermitian positive matrix ALA. The inversion can be effected by various classical algorithms, either directly (method of Gauss, Cholesky, etc.) or by an iterative technique (Gauss-Seidel algorithm, gradient algorithm, etc.).
The estimation error D−{circumflex over (D)}ZF is equal to (AHA)−1AHYN, hence the solution includes noise with a variance:σ2=E(∥D−{circumflex over (D)}ZF∥2)=N0×Trace[(AHA)−1]  (6) where N0 is the noise power spectral density. It can be seen that noise enhancement occurs when the matrix AHA is badly conditioned, i.e. when it has one or more eigenvalues close to zero.
This noise enhancement is the main drawback of the conventional solution methods. In practice, the cases where the matrix AHA is badly conditioned are frequent, especially with multiple propagation paths.
A relatively simple means to partly remedy this drawback is known, by accepting residual interference in the solution, i.e. by not adopting the optimum least-squares solution, but the solution: DMMSE=(AHA+{circumflex over (N)}0)−AHY, where {circumflex over (N)}0 denotes an estimation of the noise spectral density, that the receiver must then calculate. This is known as the MMSE (minimum mean square error) method and it allows the estimation variance to be reduced relative to the zero-forcing method, but introduces a bias.
The zero forcing methods and the like amount to performing an inverse filtering on the signal received by a filter which models the transfer function 1/R(Z) calculated using a certain approximation (quadratic in the case of zero forcing). Where one or more roots of the polynomial R(Z) (equation (5)) are situated on the unit circle, the theoretical inverse filter presents singularities rendering it impossible to estimate by a satisfactory approximation. In the case of the quadratic approximation, this corresponds to a divergence in the error variance σ2 when the matrix AHA has an eigenvalue equal to zero (relation (6)).
This problem does not arise in methods such as the Viterbi algorithm which intrinsically take the discrete nature of the symbols into account, but which require a much higher calculating power for large-sized systems.
U.S. Pat. No. 4,701,936 discloses a channel equalizer using an all-pass filter determined with reference to the Z-transform of the estimated channel impulse response.
An object of the present invention is to propose an equalization method achieving a good compromise between the reliability of the estimations and the complexity of the equalizer.
Another object is to produce an equalizer requiring a reasonable calculating power but capable of processing, with performance comparable to that of Viterbi equalizers, signals whose symbols have a relatively high number of states and/or signals carried on a channel with a relatively log impulse response.